Tag Archives: 3-manifolds

Lattice Homology and Knots

I’m excited to be joining the blog. I’ve been consumed mainly by watching talks and meeting people at the Stanford conference, so this first post will consist only of my notes from Andras Stipsicz‘s talk on lattice homology. The content of the talk represents joint work of Stipsicz with Ozsváth and Szabó. I don’t have a lot of experience with lattice homology, so if I’ve missed or misunderstood something, please contribute your corrections in the comments section.

The talk began with a review of the standard Heegaard Floer package. Knot Floer homology was recently mentioned on the blog when Laura discussed Zoltan Szabó’s research talk at Budapest. So, if you’d like a refresher on notation, please check out that post (but feel free to skip the bordered part if you’re in a rush).

With an powerful invariant like Heegaard Floer (or knot Heggaard Floer), a lot of people are interested in how to compute it. Stipsicz gave an overview of some of the pertinent questions about computation and existing technology.

Computational aspects:

  • How to compute the invariant in a simple way?
  • How to prove invariance in a simple way? (Simple here means without infinite dimensional analysis or having to count pseduoholomorphic curves.)
  • Can you find an effective way of computing invariants? (Effective depends on your personal opinion of an efficient algorithm.)

Existing technology:

The first two references mentioned are examples of simple ways to compute the invariants. The cost of theoretical simplicity, though, is a miserably inefficient algorithm. For example, the algorithm that computes HFK from a grid diagram has order of complexity somewhere in the neighborhood of O(n^2!). Yikes! The bordered techniques are computationally better, but require some rather difficult algebra.

So keeping these motivating issues in mind, let’s see what Stipsicz said about lattice homology and how it relates to Heegaard Floer.

Lattice homology
Suppose that G is a tree with integer weights m_i associated to its vertices v\in V(G). We consider the plumbing construction G \rightsquigarrow X_G. This means that X_G is the compact oriented 4-manifold obtained by plumbing D^2 bundles over S^2 with Euler numbers given by the weights m_i and plumbing instructions indicated by the graph. The three dimensional boundary Y_G = \partial X_G of the resulting manifold is our object of interest.

The construction that follows is due to Nèmethi in 2008:

Define Char(G) to be \{K:V\rightarrow \mathbb{Z} | K(v)=m_v mod 2 for all v\} , i.e. the set of characteristic elements in the second integral cohomology group of X_G.

Let \mathbb{CF}^-(G) denote the \mathbb{Z}_2[U] module freely generated over \mathbb{Z}_2[U] by pairs [K, E], where K is a characteristic covector and E is a subset of vertices in V.
The differential of this complex is given by:

\partial^-[K,E] = \Sigma_{v\in E} U^{a_v[K,E]} [K,E-v] + U^{b_v[K,E]} [K+2v^*,E-v]

Those exponents a_v and b_v are integers which are defined with an auxiliary function \mathcal{I}. If I\subset E, then \mathcal{I}[K,I] = \frac{1}{2}(\sum_{u\in I}K(u) + (\sum_{u\in I}u)^2 ). That second term represents a self-intersection in X_G. Let:

A_v[K,E] = \min\{ \mathcal{I} | I\subset E-v\}
B_v[K,E] = \min\{ \mathcal{I} | v\in I\subset E\}
a_v = A_v- \min \{A_v, B_v\}
b_v = B_v - \min \{ A_v, B_v \}

Also note that since K specifies a spin-c structure on X_G, we associate to K the spin-c structure \mathfrak{s}_K \in Spin^c(Y_G).

OK. That sets up the lattice homology, more or less. I don’t really understand what the differential is actually doing, but its apparent benefit is that it is defined in a combinatorial manner (i.e. no counts of holomorphic curves). If anyone would like to really understand this, it would probably be best to check out the papers by Nèmethi and Ozsváth, Stipsicz, and Szabó.

Rational graphs
My notes became somewhat illegible at this point in the lecture, so I’ve attempted to vet this section with some information from this paper. Let’s assume that G is a negative definite plumbing graph. We’d like to get a handle on when G is rational, and as it turns out, there is a combinatorial algorithm that does this.

Algorithm. Let K_1 = \Sigma_{v\in V} v, and compute all of the products K_1\cdot v. If:

  • any of the products is greater than or equal to 2, stop. G is not rational.
  • all of the products are less than or equal to 0, stop. G is rational.
  • any product K_1\cdot v = 1, then set K_2 = K_1+ v and repeat.

(I’m being a bit sloppy here with the notation; v stands for both a vertex and the second homology class in the plumbed manifold corresponding to that vertex.) This algorithm produces a series of vectors K_1, K_2, \cdots, and terminates in finite time. The product, by the way, is a dot product computed in the intersection matrix of the four-manifold. I believe E_8 was a suggested example of a rational graph on which to perform the algorithm.

Additionally, we say that G is of type-k if there exists vertices v_1, \cdots, v_k such that sufficiently decreasing the weights m_{v_1}, \cdots m{v_k} makes G' rational.

So why do we care about types of rational graphs? The following theorem of Nèmethi in 2008:
Theorem. Suppose G is negative definite plumbing tree. Then,

  • H_*( \mathbb{CF}^-(G, \mathfrak{s}), \partial^-) =\mathbb{HF}^-(G, \mathfrak{s}) is a three-manifold invariant.
  • G is rational if and only if \mathbb{HF}^-(G, \mathfrak{s}) \cong \mathbb{Z}_2[U] for all spin-c structures over Y_G.
  • If G is of type-1 this implies \mathbb{HF}^-(G, \mathfrak{s}) \cong HF^-(Y_G, \mathfrak{s}).

Moreover, there exists a surgery exact sequence for lattice homology, a result also obtained independently by Josh Greene. The theorem leads to the conjecture that \mathbb{HF}^-(G, \mathfrak{s}) \cong HF^-(Y_G, \mathfrak{s}) for any plumbing tree G. To this end, there is a theorem of Ozsváth, Stipsicz, and Szabó that tells us there is a spectral sequence on \mathbb{HF}^-(G, \mathfrak{s}) converging to HF^-(Y_G, \mathfrak{s}). If G is of type-2, the spectral sequence collapses and the homology groups are isomorphic.

Refinement to knots

Are you still hanging in there? Great! Onward to knots.

Now, we let \Gamma_{v_0} denote a plumbing graph with a distinguished, unweighted vertex v_0. Define the lattice chain complex on G=\Gamma_{v_0}-v_0. The vertexv_0 induces a filtration \mathcal{A} on the lattice homology complex. For a generator [K, E] of the lattice complex, the filtration level \mathcal{A}[K,E] is given by a formula that is essentially the same as the auxiliary function \mathcal{I} above, except it use a specific extension of K and E to \Gamma_{v_0}. The filtered complex (\mathbb{CF}^-(G, \mathfrak{s}), \partial^-, \mathcal{A}_{v_0}) is the subject of the main theorem:

Theorem. Suppose the tree \Gamma_{v_0} with distinguished vertex v_0 is given, that G=\Gamma_{v_0}-v_0 negative definite, and that G_{v_0}(k) is negative definite* for k\in \mathbb{Z}. Then \mathbb{CF}^-(G, \partial^-, \mathcal{A}_{v_0}) determines the lattice chain complex \mathbb{CF}^-(G_{v_0}(k), \partial^-).

(*I took G_{v_0}(k) to mean that we’ve put a framing of k on the vertex v_0, large enough to ensure that Y_{G_{v_0}} is an L-space.)

You might be asking yourself where does the knot fit into this picture? Well, when you do the plumbing, the unknot living at the distinguished vertex becomes knotted. That’s the knot. It’s also important to note that while knot Floer homology is defined for all knots, the construction described above is not. It works for connected sums of iterated torus knots, but certainly not all knots.

I do not have very detailed notes about the proof of the theorem, but it seemed to me to be a consequence of a complicated and formal similarity of homological algebra. In Heegaard Floer, CF^\infty is a doubly filtered complex. The filtrations can be used to chop up the complex into various subcomplexes, and then these these subcomplexes, the maps relating them, and their mapping cones are used to describe the Heegard Floer of surgeries along K. Here is a paper that details this construction.

Apparently, lattice homology also enjoys such a structure. The ‘infinity’ lattice complex \mathbb{CF}^\infty(G) = \mathbb{CF}^-(G)\otimes_{\mathbb{Z}_2} \mathbb{Z}_2[U^{-1}, U] is doubly filtered by the action of U and an extension of \mathcal{A}. It, too, can be chopped up into subcomplexes, the mapping cones of which are quasi-isomorphic to lattice complexes of G_{v_0}(k).

Relating the Heegaard Floer and lattice complexes takes more work. Unfortunately, there isn’t much I am able to say about this. So, let me just say that lattice homology is really interesting and I’m looking forward to learning more about it.

I’ll sign-off by reporting the big result in Ozsváth, Stipsicz, and Szabó‘s paper.
Theorem. If G is a plumbing graph with a distinguished vertex v such that the components of G-v are all rational, then the filtered chain complex of \Gamma_{v_0} in lattice homology is filtered chain homotopy equivalent to the filtered complex for Y_G in Heegaard Floer.

…and it’s wonderful corollary:
Corollary. The lattice homology of a knot in S^3 is equal to its knot Floer homology.

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Live from Budapest

From July 9-20, we’re at the CAST (Contact and Symplectic Topology) 2012 summer school and conference at the Alfred Renyi Institute of Mathematics in Budapest, Hungary.  The format is two mini-courses per week along with two research talks every afternoon.

First Week:

Robert Lipshitz – Bordered Heegaard Floer Homology (lecture notes)

This week, Robert lectured on a Heegaard Floer-like structure associated to bordered 3-manifolds, which was constructed by himself, Peter Ozsvath and Dylan Thurston.  This associates two modules, \hat{CFA}(Y), \hat{CFD}(Y) to a 3-manifold with connected boundary, depending upon whether we think of the boundary as positively or negatively oriented.  There are also more complicated modules associated to the nonconnected boundary case that mixes -A and -D structure.  The main theorem is the Pairing Theorem (LOT), which states that for Y = Y_1 \cup Y_2 then \hat{CF}(Y) = \hat{CFA}(Y_1) \otimes \hat{CFD}(Y_2) where \hat{CF} is the chain complex  for the hat version of Heegaard Floer.  For the first three days, he developed the terminology and combinatorics of parametrizations of the boundary \partial Y, defined the differential and discussed the relevant moduli of holomorphic curves.  One key aspect of this construction is that it relies on his cylindrical reformulation of HF to bring it more in line analytically with Symplectic Field Theory.  The fourth day, Jen Hom went over how to compute \hat{CFA}(Y) for knot complements and the fifth day Robert discussed computing \hat{CFA}(Y) for mapping tori and that it may be possible to simplify computations of \hat{HF}(Y) by computing the bordered version for generators of the mapping class group and composition of these elementary cobordisms.

Kai Cieliebak – Stein Structures: Existence and Flexibility (lecture notes)

The subject of this talk was building Stein structures and classifying them up to the appropriate homotopy equivalence.  Most of the content comes from an upcoming textbook on Stein structures written by Cieliebak and Yasha Eliashberg.  A Stein manifold is a complex manifold that embeds properly, holomorphically into some \mathbb{C}^n; equivalently, it admits an exhausting (proper, bounded below), J-convex (or strictly plurisubharmonic) function \phi, which can be used to set up the embedding into affine space.  An important result, which is due to Milnor and can be verified easily, is that the Morse index of a nondegenerate critical point of \phi must be less than or equal to n.  The goal of the lecture series is to prove the converse, that if an open, smooth, oriented manifold M of dimension 2n > 4 admits an almost complex structure J and a generic Morse function \phi with critical points of index less than n, then there is a homotopy of almost-complex structures from J to some J' so that \phi is J'-convex, giving a Stein structure.  The basic idea is to use a Stein h-cobordism theorem to simplify the set of critical points and attaching spheres of handles, build a model J-convex structure on a handles and extend the standard Stein structure on the unit ball as we add on handles.  The last issue is flexibility and classifying Stein structures up to Stein homotopy.  In the subcritical case, it follows from the h-principle for isotropic embeddings that if two Stein structures (V,J), (V',J') have homotopic almost-complex structures then they are Stein homotopic.  This is enough for the subcritical case, when M has the homotopy type of an n-1-dimensional manifold.  The critical case relies on Max Murphy’s notion of loose legendrians, which do satisfy an h-principle.   A Stein structure is flexible if its critical attaching spheres are loose, and flexible Stein structures with homotopic almost-complex structures are in fact Stein homotopic.

Second Week:

Michael Hutchings – Embedded Contact Homology

Lenny Ng – Knot Contact Homology

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